Boundary Conditions and Range of Force forSState of Two Protons

The paper is divided into five sections the first of which is an introduction. In the second the possibilities of describing phase shifts by means of a boundary condition at a distance small compared with $\frac{e^2}{m{c}^2}=2.8\mathrm{}\times {10}^{-13\mathrm{}}$ cm is discussed. It is brought out that the irregular solution of the wave equation in a Coulomb field has a logarithmic infinity which masks the features of the wave function which have to do with phase shifts and therefore with observation. In Table I approximate values of essential quantities are listed. In Section III boundary conditions at moderate distances are studied. It is found that one can replace the "potential energy curve" description by the requirement that the logarithmic derivative of the wave function have an energy independent value at a distance of $\sim 0.47\frac{e^2}{m{c}^2}$. Similarly the ${}_{}{}^1{}_{}{}^{}S$ proton-neutron interaction can be approximately described by requiring the logarithmic derivative to have an energy independent value at $\sim 0.49\frac{e^2}{m{c}^2}$. In the convention of dealing with distance times radial function the values of the logarithmic derivatives are ∼0.08, 0.06 for the proton and neutron cases, respectively. It is also possible to require a linear variation of energy for the logarithmic derivative within limits and to retain agreement with experiment. It is pointed out in the introduction that theoretical arguments for considering a failure of the potential energy viewpoint exist and that the agreement of the boundary conditions of Section III with observation may be more than an accident. In Section IV the adjustment of the range of force is treated and evidence for a somewhat smaller value than 2.8×${10}^{-13\mathrm{}}$ cm, perhaps 2.6×${10}^{-13\mathrm{}}$ cm is discussed. Use is made of simple relationships between effective depth variation with energy and range. In Section V the function $f$ of BCP is expanded in powers of energy $E$, the relations for potential energy curves of different shapes are taken up regarding equivalence of range, the deviations from linearity of $f$ with $E$ are discussed from the viewpoint of equivalent error in scattering, and a rapid procedure for finding the equivalent square well range by means of successive approximations is given.